Free-energy Transducers Research Articles

Gear Shifting in Biological Energy Transduction.

Confronted with thermodynamically adverse output processes, free-energy transducers may shift to lower gears, thereby reducing output per unit input. This option is well known for inanimate machines such as automobiles, but unappreciated in biology. The present study extends existing non-equilibrium thermodynamic principles to underpin biological gear shifting and identify possible mechanisms. It shows that gear shifting differs from altering the degree of coupling and that living systems may use it to optimize their performance: microbial growth is ultimately powered by the Gibbs energy of catabolism, which is partially transformed into Gibbs energy ('output force') in the ATP that is produced. If this output force is high, the cell may turn to a catabolic pathway with a lower ATP stoichiometry. Notwithstanding the reduced stoichiometry, the ATP synthesis flux may then actually increase as compared to that in a system without gear shift, in which growth might come to a halt. A 'variomatic' gear switching strategy should be optimal, explaining why organisms avail themselves of multiple catabolic pathways, as these enable them to shift gears when the growing gets tough.

Theory of Nonequilibrium Free Energy Transduction by Molecular Machines.

Biomolecular machines are protein complexes that convert between different forms of free energy. They are utilized in nature to accomplish many cellular tasks. As isothermal nonequilibrium stochastic objects at low Reynolds number, they face a distinct set of challenges compared with more familiar human-engineered macroscopic machines. Here we review central questions in their performance as free energy transducers, outline theoretical and modeling approaches to understand these questions, identify both physical limits on their operational characteristics and design principles for improving performance, and discuss emerging areas of research.

Stochastic Dynamics of Proteins and the Action of Biological Molecular Machines

It is now well established that most if not all enzymatic proteins display a slow stochastic dynamics of transitions between a variety of conformational substates composing their native state. A hypothesis is stated that the protein conformational transition networks, as just as higher-level biological networks, the protein interaction network, and the metabolic network, have evolved in the process of self-organized criticality. Here, the criticality means that all the three classes of networks are scale-free and, moreover, display a transition from the fractal organization on a small length-scale to the small-world organization on the large length-scale. Good mathematical models of such networks are stochastic critical branching trees extended by long-range shortcuts. Biological molecular machines are proteins that operate under isothermal conditions and hence are referred to as free energy transducers. They can be formally considered as enzymes that simultaneously catalyze two chemical reactions: the free energy-donating (input) reaction and the free energy-accepting (output) one. The far-from-equilibrium degree of coupling between the output and the input reaction fluxes have been studied both theoretically and by means of the Monte Carlo simulations on model networks. For single input and output gates the degree of coupling cannot exceed unity. Study simulations of random walks on model networks involving more extended gates indicate that the case of the degree of coupling value higher than one is realized on the mentioned above critical branching trees extended by long-range shortcuts.

Output-input ratio in thermally fluctuating biomolecular machines

Biological molecular machines are proteins that operate under isothermal conditions and hence are referred to as free energy transducers. They can be formally considered as enzymes that simultaneously catalyze two chemical reactions: the free energy-donating (input) reaction and the free energy-accepting (output) one. Most if not all biologically active proteins display a slow stochastic dynamics of transitions between a variety of conformational substates composing their native state. This makes the description of the enzymatic reaction kinetics in terms of conventional rate constants insufficient. In the steady state, upon taking advantage of the assumption that each reaction proceeds through a single pair (the gate) of transition conformational substates of the enzyme-substrates complex, the degree of coupling between the output and the input reaction fluxes has been expressed in terms of the mean first-passage times on a conformational transition network between the distinguished substates. The theory is confronted with the results of random-walk simulations on the five-dimensional hypercube. The formal proof is given that, for single input and output gates, the output-input degree of coupling cannot exceed unity. As some experiments suggest such exceeding, looking for the conditions for increasing the degree of coupling value over unity challenges the theory. Performed simulations of random walks on several model networks involving more extended gates indicate that the case of the degree of coupling value higher than 1 is realized in a natural way on critical branching trees extended by long-range shortcuts. Such networks are scale-free and display the property of the small world. For short-range shortcuts, the networks are scale-free and fractal, representing a reasonable model for biomolecular machines displaying tight coupling, i.e., the degree of coupling equal exactly to unity. A hypothesis is stated that the protein conformational transition networks, as just as higher-level biological networks, the protein interaction network, and the metabolic network, have evolved in the process of self-organized criticality.

Thermodynamics of biochemical networks and duality theorems

One interesting yet difficult computational issue has recently been posed in biophysics in regard to the implementation of thermodynamic constraints to complex networks. Biochemical networks of enzymes inside cells are among the most efficient, robust, differentiated, and flexible free-energy transducers in nature. How is the second law of thermodynamics encoded for these complex networks? In this article it is demonstrated that for chemical reaction networks in the steady state the exclusion (presence) of closed reaction cycles makes possible (impossible) the definition of a chemical potential vector. Interestingly, this statement is encoded in one of the key results in combinatorial optimization, i.e., the Gordan theorem of the alternatives. From a computational viewpoint, the theorem reveals that calculating a reaction's free energy and identifying infeasible loops in flux states are dual problems whose solutions are mutually exclusive, and this opens the way for efficient and scalable methods to perform the energy balance analysis of large-scale biochemical networks.

Comment: Detailed balance revisited

The concept of detailed balance is central to the field of chemical kinetics and provides important constraints on relationships between equilibrium and rate constants. Miller et al. [Phys. Chem. Chem. Phys., 2009, 11, 1128] have recently proposed that having the ratio of forward and reverse rate constants equal to the equilibrium constant--a relationship they equate with detailed balance--is a sufficient condition to guarantee that a chemical system eventually relaxes to equilibrium. This assertion is true, however, only for an isolated system. In a closed system, where energy exchange between the system and the environment occurs, input of energy in the form of an externally driven oscillation of a thermodynamic parameter (e.g. pressure or electric field strength), can maintain a chemical system in a stationary state away from equilibrium even if the ratio of the forward and backward rate constants for each elementary reaction equals the equilibrium constant at every instant. This fact allows appropriately designed molecules to serve as free-energy transducers and molecular machines.

Variation of efficiency with free-energy dissipation in models of biological energy transduction

For two models of biological free-energy transducers, it is investigated how free-energy dissipation and efficiency vary as (i) the demand for output free energy, (ii) the input free energy or (iii) the properties of the transducers themselves, are varied. One model is representative of near-equilibrium free-energy transducers in general, the other is a special case of far-from-equilibrium free-energy transduction, reminiscent of proton pumping by bacteriorhodopsin. It turns out that the relationship between efficiency and free-energy dissipation depends strongly on what varies. In some cases, free-energy dissipation increases as the efficiency increases. It is suggested that this is one reason why biological evolution has not resulted in high efficiencies and low rates of free-energy dissipation. For the near-equilibrium free-energy transducer, the free-energy dissipation at the static head steady state is minimal with respect to variations in the output force. For the far-from-equilibrium model (of bacteriorhodopsin), the static head does not correspond to such a minimum, if that free-energy transducer slips.

Twenty Questions Concerning The Reaction Cycle Of The Sarcoplasmic Reticulum Calcium Pum

The problem of "mechanism" for the calcium pump may be divided into three parts. (1) It is an enzyme catalyzing the hydrolysis of ATP. (2) At some stage of the reaction cycle it provides a pathway through the otherwise impermeable phospholipid bilayer. (3) The two properties are linked so as to provide exchange of free energy between the two substrates (ATP and Ca2+), without any exchange of matter. The third part is the most interesting, and the mechanistic problem it poses is common to all chemiosmotic free energy transducers. All three aspects of the mechanism are reviewed here, with special emphasis on the remaining experimental questions that need to be resolved. The review will show that even such fundamental questions as the exact stoichiometry of the catalyzed reaction have not yet received definitive answers.